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Maximum Matching on graphs in the semi-streaming model

Maximum Matching in simple undirected graphs is a subset of maximum number of edges such that at most one edge is incident on any vertex. We study this problem in the semi-streaming model, in which edges are presented as a stream, and the goal is to output a matching of maximum size while only using $O(n\polylog n)$ space (where $n$ is the number of vertices in the graph). A $k$-pass algorithm can go over the stream $k$ times.

In this work, we present a two pass algorithm which produces a matching of size at least (1/2+1/16) times that of maximum matching. We also present a $2/(3\epsilon)$-pass algorithm which gives a matching of size at least $2/3-\epsilon$ times that of maximum matching.

Abstract

In the online bipartite matching problem with replacements, all the vertices
on one side of the bipartition are given, and the vertices on the other side
arrive one by one with all their incident edges. The goal is to maintain a
maximum matching while minimizing the number of changes (replacements) to the
matching. We show that the greedy algorithm that always takes the shortest
augmenting path from the newly inserted vertex (denoted the SAP protocol) uses
at most amortized $O(\log^2 n)$ replacements per insertion, where $n$ is the
total number of vertices inserted. This is the first analysis to achieve a
polylogarithmic number of replacements for \emph{any} replacement strategy,
almost matching the $\Omega(\log n)$ lower bound. The previous best known
strategy achieved amortized $O(\sqrt{n})$ replacements [Bosek, Leniowski,
Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better
than then trivial $O(n)$ bound was known except in special cases.
Our analysis immediately implies the same upper bound of $O(\log^2 n)$
reassignments for the capacitated assignment problem, where each vertex on the
static side of the bipartition is initialized with the capacity to serve a
number of vertices.
We also analyze the problem of minimizing the maximum server load. We show
that if the final graph has maximum server load $L$, then the SAP protocol
makes amortized $O( \min\{L \log^2 n , \sqrt{n}\log n\})$ reassignments. We
also show that this is close to tight because $\Omega(\min\{L, \sqrt{n}\})$
reassignments can be necessary.